DE-BUNDESWETTBEWERB 2000

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$ . Is this possible when each of the numbers consists of $2000$ digits?

A convex quadrilateral $ABCD$ is inscribed in a semicircle with diameter $AB$ . The diagonals $AC,BD$ intersect at $S$ , and $T$ is the projection of $S$ on $AB$ . Show that $ST$ bisects angle $CTD$ .

We are given $n \geq 3$ weights of masses $1, 2, 3, \ldots , n$ grams. Find all $n$ for which it is possible to divide these weights into three groups with the same mass.

Prove that for every integer $n \geq 2$ there exist $n$ different positive integers such that for any two of these integers $a$ and $b$ their sum $a+b$ is divisible by their difference $a - b.$

Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if**(a)** $n=2000;$ (b) $n=2001 \ ?$